Existence of mild solutions and regularity criteria of weak solutions to the viscoelastic Navier–Stokes equation with damping

In this paper, we consider the viscoelastic Navier–Stokes equation (VNS) with damping in the whole space. We first show that there exist global mild solutions with small initial data in the scaling invariant space. The main technique we have used is implicit function theorem which yields necessarily continuous dependence of solutions on the initial data. Moreover, we derive the asymptotic stability of solutions as the time goes to infinity. As a byproduct of our construction of solutions in the weak $L^p$-spaces, the existence of self-similar solutions was established provided the initial data are small homogeneous functions. Next, we deduce the regularity criteria of weak solutions to VNS with damping. Sufficient conditions for the regularity of weak solutions are presented by imposing Serrin’s‑type growth conditions on the velocity field and deformation tensor in Lorentz spaces, multiplier spaces, bounded mean oscillation spaces and Besov spaces, respectively.

Keywords

viscoelastic Navier–Stokes equation, mild solutions, asymptotic stability, self-similar solutions, regularity criteria

2010 Mathematics Subject Classification

35Q30, 35Q35, 76A10, 76B03, 76D03

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The first and second authors were supported by the National Natural Science Foundation of China (No.11726023, 11531010). The third author was supported by the Postdoctoral Science Foundation of China (No. 2019TQ0006).

Received 27 September 2018

Accepted 15 September 2019

Published 1 April 2020